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**The first case of Fermat’s last theorem is true for all prime exponents up to \(714,591,416,091,389\).**
*(English)*
Zbl 0645.10018

This is one of the most important papers on the first case of Fermat’s last theorem. In 1917, Pollaczek extended previous results of Wieferich, Mirimanoff, Vandiver, Frobenius, and showed that if there exist integers x, y, z, not multiples of the prime p and satisfying \(x^ p+y^ p=z^ p\) (i.e., if the first case of Fermat’s last theorem is false for the exponent p) then \(\ell^{p-1}\equiv 1(mod p^ 2)\) for every prime \(\ell \leq 31\). Actually, Pollaczek’s proof seemed only to hold for \(p>\alpha^{\ell^ 2/3}\), where \(\alpha =(1+\sqrt{5})/2\). Later results of Morishima and Rosser (for \(\ell =37\), 41, 43), based on Pollaczek’s claim, were therefore questionable. Up to this time, it has been unclear how to fix Pollaczek’s work beyond any doubt. It was also very enticing to attempt to extend the results for large primes \(\ell.\)

The very complex - and clever - method of Pollaczek had to be clarified and streamlined; the calculations, involving very large determinants with polynomials entries, and the computation of resultants, as well as the determination of their factors - all this offered a challenge, not at all easy to meet.

It is the primary merit of the authors to have succeeded in this program. The starting point is an analysis of the Kummer-Mirimanoff congruences. A thorough algebraic preparation, concerned with power series, serves to put in solid ground the subsequent developments. A crucial step in the studies of Pollaczek and Morishima concerned \(t\equiv x/y (mod p.)\) Rectifying their previous mistakes, Gunderson established that t cannot have order 3, 4 or 6 (mod p).

This fact is needed in an inductive process first devised by Frobenius, which would allow to show the congruence \(\ell^{p-1}\equiv 1\) (mod \(p^ 2)\) for increasingly larger primes \(\ell\). All this, provided some determinants and resultants could be effectively evaluated. This major problem was handled with success by the authors, who conceived clever tricks, using symmetries inherent to the problem, in order to reduce the cost of the computations. Factorizations of the very large resultants could only be performed with the recent elliptic curve algorithm.

In principle, the method might be pushed further, however cost and time were forbidding. According to a theoretical estimate of Gunderson’s function by Shanks and Williams, it follows from all the criteria \(\ell^{p-1}\equiv 1\) (mod \(p^ 2)\), for \(\ell \leq 89\), that if the first case of Fermat’s last theorem is false for p, then \(p>714,591,416,091,389.\) As added in proof, Tanner and Wagstaff have improved the estimates in Gunderson’s function, and this implies that \(p>156,442,236,847,241,650.\) This is a record for the first case. It is worth adding here that L. M. Adleman, D. R. Heath-Brown and E. Fouvry have shown [Invent. Math. 79, 409-416 (1985; Zbl 0557.10034); ibid. 79, 383-407 (1985; Zbl 0557.10035)] that the first case of Fermat’s last theorem is true for infinitely many prime exponents.

The very complex - and clever - method of Pollaczek had to be clarified and streamlined; the calculations, involving very large determinants with polynomials entries, and the computation of resultants, as well as the determination of their factors - all this offered a challenge, not at all easy to meet.

It is the primary merit of the authors to have succeeded in this program. The starting point is an analysis of the Kummer-Mirimanoff congruences. A thorough algebraic preparation, concerned with power series, serves to put in solid ground the subsequent developments. A crucial step in the studies of Pollaczek and Morishima concerned \(t\equiv x/y (mod p.)\) Rectifying their previous mistakes, Gunderson established that t cannot have order 3, 4 or 6 (mod p).

This fact is needed in an inductive process first devised by Frobenius, which would allow to show the congruence \(\ell^{p-1}\equiv 1\) (mod \(p^ 2)\) for increasingly larger primes \(\ell\). All this, provided some determinants and resultants could be effectively evaluated. This major problem was handled with success by the authors, who conceived clever tricks, using symmetries inherent to the problem, in order to reduce the cost of the computations. Factorizations of the very large resultants could only be performed with the recent elliptic curve algorithm.

In principle, the method might be pushed further, however cost and time were forbidding. According to a theoretical estimate of Gunderson’s function by Shanks and Williams, it follows from all the criteria \(\ell^{p-1}\equiv 1\) (mod \(p^ 2)\), for \(\ell \leq 89\), that if the first case of Fermat’s last theorem is false for p, then \(p>714,591,416,091,389.\) As added in proof, Tanner and Wagstaff have improved the estimates in Gunderson’s function, and this implies that \(p>156,442,236,847,241,650.\) This is a record for the first case. It is worth adding here that L. M. Adleman, D. R. Heath-Brown and E. Fouvry have shown [Invent. Math. 79, 409-416 (1985; Zbl 0557.10034); ibid. 79, 383-407 (1985; Zbl 0557.10035)] that the first case of Fermat’s last theorem is true for infinitely many prime exponents.

Reviewer: P.Ribenboim

### MSC:

11D41 | Higher degree equations; Fermat’s equation |